Kinematics Problem and stream of people

1. The problem statement, all variables and given/known data
Figure 2-21 shows a general situation
in which a stream of people attempt to escape through an exit door
that turns out to be locked. The people move toward the door at
speed vs 3.50 m/s, are each d 0.25 m in depth, and are separated
by L 1.75 m.The arrangement in Fig. 2-21 occurs at time t 0. (a)
At what average rate does the layer of people at the door increase?
(b) At what time does the layer’s depth reach 5.0 m? (The answers
reveal how quickly such a situation becomes dangerous.)

Staff: Mentor

1. The problem statement, all variables and given/known data
Figure 2-21 shows a general situation
in which a stream of people attempt to escape through an exit door
that turns out to be locked. The people move toward the door at
speed vs 3.50 m/s, are each d 0.25 m in depth, and are separated
by L 1.75 m.The arrangement in Fig. 2-21 occurs at time t 0. (a)
At what average rate does the layer of people at the door increase?
(b) At what time does the layer’s depth reach 5.0 m? (The answers
reveal how quickly such a situation becomes dangerous.)

2. Relevant equations

3. The attempt at a solution
I am not certain how to solve this.

What equations in general do you think are applicable?

How long do you think the time interval will be between successive impacts?

So, in 1 second the first person collides into the door, and then the second person collides into the first person. Since each person has a depth of .25 m, that would mean the depth at the door would increase .5 m/s? That doesn't seem right. Isn't the depth at the door increases exponentially?

Staff: Mentor

So, in 1 second the first person collides into the door, and then the second person collides into the first person. Since each person has a depth of .25 m, that would mean the depth at the door would increase .5 m/s? That doesn't seem right. Isn't the depth at the door increases exponentially?

As you stated earlier, the first person hits the door in 0.5 seconds, not 1 second. After that they "pile on" at a rate of one every 0.5 seconds. At the moment you're interested in this "after that" scenario, since it represents the ongoing rate of accumulation of people on the pile.

Note that there is no exponential increase in the rate that people arrive --- they arrive at a constant rate, so the depth is increasing at an average constant rate, too: At 0.5m/sec in fact, as you've discovered.